CK-12 Geometry - Second Edition

http://www.ck12.org Chapter 7. Similarity

a)^45 = 30 x

b)y+ 81 = 205

c)^65 =^2 xx−+ 25

Solution:To solve a proportion, you need tocross-multiply.

a)

b)

c)

In proportions, the blue numbers are called themeansand the orange numbers are called theextremes. For the
proportion to be true, the product of the means must equal the product of the extremes. This can be generalized in
the Cross-Multiplication Theorem.

Cross-Multiplication Theorem:Leta,b,c,anddbe real numbers, withb 6 =0 andd 6 =0. Ifab=cd, thenad=bc.

The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by^22 ,bb,ordd= 1
because it is the same number divided by itself(b÷b= 1 ).

Proof of the Cross-Multiplication Theorem

a

b

=

c

d

Multiply the left side by

d

d

and the right side by

b

b

.

a

b

·

d

d

=

c

d

·

b

b

ad

bd

=

bc

bd

The denominators are the same, so the numerators are equal.

ad=bc

Think of the Cross-Multiplication Theorem as a shortcut. Without this theorem, you would have to go through all of
these steps every time to solve a proportion.